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Studying for a test? Prepare with these 7 lessons on Linear equations and inequalities.
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We have the proportion x minus 9 over 12 is equal to 2/3. And we want to solve for the x that satisfies this proportion. Now, there's a bunch of ways that you could do it. A lot of people, as soon as they see a proportion like this, they want to cross multiply. They want to say, hey, 3 times x minus 9 is going to be equal to 2 times 12. And that's completely legitimate. You would get-- let me write that down. So 3 times x minus 9 is equal to 2 times 12. So it would be equal to 2 times 12. And then you can distribute the 3. You'd get 3x minus 27 is equal to 24. And then you could add 27 to both sides, and you would get-- let me actually do that. So let me add 27 to both sides. And we are left with 3x is equal to-- let's see, 51. And then x would be equal to 17. And you can verify that this works. 17 minus 9 is 8. 8/12 is the same thing as 2/3. So this checks out. Another way you could do that, instead of just straight up doing the cross multiplication , you could say look, I want to get rid of this 12 in the denominator right over here. Let's multiply both sides by 12. So if you multiply both sides by 12, on your left-hand side, you are just left with x minus 9. And on your right-hand side, 2/3 times 12. Well, 2/3 of 12 is just 8. And you could do the actual multiplication. 2/3 times 12/1. 12 and 3, so 12 divided by 3 is 4. 3 divided by 3 is 1. So it becomes 2 times 4/1, which is just 8. And then you add 9 to both sides. So the fun of algebra is that as long as you do something that's logically consistent, you will get the right answer. There's no one way of doing it. So here you get x is equal to 17 again. And you could also-- you could multiply both sides by 12 and both sides by 3, and then that would be functionally equivalent to cross multiplying. Let's do one more. So here another proportion. And this time the x in the denominator. But just like before, if we want we can cross multiply. And just to see where cross multiplying comes from, it's not some voodoo, that you still are doing logical algebra, that you're doing the same thing to both sides of the equation, you just need to appreciate that we're just multiplying both sides by both denominators. So we have this 8 right over here on the left-hand side. If we want to get rid of this 8 on the left-hand side in the denominator, we can multiply the left-hand side by 8. But in order for the equality to hold true, I can't do something to just one side. I have to do it to both sides. Similarly, if I want to get this x plus 1 out of the denominator, I could multiply by x plus 1 right over here. But I have to do that on both sides if I want my equality to hold true. And notice, when you do what we just did, this is going to be equivalent to cross multiplying. Because these 8's cancel out, and this x plus 1 cancels with that x plus 1 right over there. And you are left with x plus 1 times 7-- and I can write it as 7 times x plus 1-- is equal to 5 times 8. Notice, this is exactly what you would have done if you had cross multiplied. Cross multiplication is just a shortcut of multiplying both sides by both the denominators. We have 7 times x plus 1 is equal to 5 times 8. And now we can go and solve the algebra. So distributing the 7, we get 7x plus 7 is equal to 40. And then subtracting 7 from both sides, so let's subtract 7 from both sides, we are left with 7x is equal to 33. Dividing both sides by 7, we are left with x is equal to 33/7. And if we want to write that as a mixed number, this is the same thing-- let's see, this is the same thing as 4 and 5/7. And we're done.